We are gonna call x the greatest number, y is the middle number and z is the smallest number.
"The greatest of three numbers is six times the smallest" is expressed as
[tex]x=6z[/tex]"The middle number is 10 more than the smallest" is expressed as
[tex]y=10+z[/tex]"Twice the greatest is decreased by 90 the result is the sum of the middle and the smallest" is expressed as
[tex]2x-90=y+z[/tex]First, we replace the first equation into the third one
[tex]\begin{gathered} 2(6z)-90=y+z \\ 12z-90=y+z \\ -90=y+z-12z \\ -90=y-11z \end{gathered}[/tex]Now, we replace the second equation into the last one above
[tex]\begin{gathered} -90=10+z-11z \\ -90-10=-10z \\ -100=-10z \\ z=\frac{-100}{-10}=10 \end{gathered}[/tex]Then, we find x
[tex]x=6(10)=60[/tex]We find y
[tex]y=10+z=10+10=20[/tex]